This disclosure is related to an improved method for searching for a minimum of a multi-dimensional surface. More, particularly, an algorithm for determining the minimum of a multi-dimensional surface used in a calibration procedure for a clutch.
The control of an electromechanical actuated rear differential unit (RDU) with one or more clutches must be robust against changing conditions, including wear of the clutches. A clutch is a mechanical device used to transfer power from one rotating shaft to another rotating shaft. When the clutch is actuated (i.e. compressed) and slipping, torque can be transferred from one shaft to the other, of which the amount depends on the normal force applied on the clutch and the level of slip. When the clutch is fully disengaged (i.e. decompressed), the shafts are decoupled and no torque is transferred. The clutch kiss point is the physical position at which the clutch its driving plates meet the clutch its driven clutch plates and starts to deliver torque. When the clutch is transferring torque the friction material of the plates will decrease. As a consequence, the amount of compression needed to engage a clutch to specific levels of transfer torque varies as the clutch wears. To improve the operation of a clutch system the system can be adapted to the unique characteristics of each clutch in order to compensate for the wear of the clutches.
To adapt to these changing conditions, methods are used to update the relationship between the applied actuator position and the applied normal force on the clutch. This relationship can be modeled by fitting a spline function to recorded data to provide as much accuracy as possible. The difficulty lies in finding the optimal spline fit. The optimal spline fit has the least amount of error with respect to the recorded data.
A multi-dimensional surface can be used to represent the error between the actual recorded data and a spline function fit with varying parameters. Each point on the multi-dimensional surface corresponds to an error value of a spline function. Minimizing the error of the spline function and thus finding the minimum of the multi-dimensional surface is the desired goal. Many of the search algorithms designed for multi-dimensional surfaces use derivatives in order to find the local or global extrema of the surface. However, running these algorithms using embedded central processing units create timing problems because calculating derivatives requires that either the exact function of the surface to be known or many iterations of the algorithms have to be executed before the derivative is known. Limiting the load placed on the central processing units and execution time of the algorithm is desired.
Additionally, the shape of the multi-dimensional surface can create issues when implementing the existing search algorithms. When the multi-dimensional surface has a large region of minimum values, variations between function values are small compared to the variations in function values closer to the outer boundaries making achieving an accurate overall minimum is difficult.
One algorithm used for determining the minimum is the brute force method which evaluates all data points within a boundary sequentially and when a new minimum value is found, the previous value is overwritten. Another advanced algorithm used for multi-dimensional surfaces, which does not use any derivatives, is the Nelder-mead (Simplex) method. This method uses a simplex (e.g. a triangle) constructed out of test points and in each iteration, the test points are evaluated and determined to expand, contract or reduce the simplex. The heuristic character of this algorithm does not always guarantee the optimal solution and an accurate minimum.
The Golden Section Search (GSS) is another known algorithm used to find extremes of two-dimensional unimodal functions within certain boundaries. The GSS algorithm uses three points whose distances form a golden section ratio. The function values of the three points are calculated and the point corresponding to the highest value is disregarded. A new point is then positioned between the two remaining points based on the golden section ratio. The known GSS algorithm is very robust and accurate in finding minima of two-dimensional lines; however, it is not used for multi-dimensional surfaces.
Therefore, a need exists for a method for determining the minimum of a multi-dimensional surface that limits the load placed on the control processing unit, reduces the execution time of the algorithm and accurately detects the minimum.